Fréchet L-Spaces

Question

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also introduced another similar class of abstract spaces called Limit spaces based on the primitive idea of the limit of an infinite sequence in 1904, which was defined as follows:

An L-space is a set $X$ together with a function $F : S\to X,$ where $S$ is a set of infinite sequences of members of $X$.

If $(x_n)\in S$, then $F(x_n)$ was said to be the “limit of the sequence $(x_n)$" satisfying following two axioms:

$A_1$: If $(x_n)$ is a constant sequence whose value is $a$, then $F(x_n)=a$.

$A_2$: If $F(x_n)=a$, then for any sub-sequence of $(x_n)$ given by $(x_{n_k})$ we have $F(x_{n_k})=a$.

I would like to know more about mathematics on L-spaces. But I could not find any thing by just Googling.
Where could I find about these spaces?

Answer 1

I think that these spaces don't go under the name of $L$ spaces anymore. Actually, I am not sure if there is a consensus on how these structures are called today.

A good place to start is the fairly recent book

• Convergence Structures and Applications to Functional Analysis by R. Beattie and H.-P. Butzmann, Kluwer, 2002

where these structures are named convergence spaces, more precisely sequential convergence spaces and are studied in chapter 1.7.

Another, slightly older, reference is the article (in French)

• Convergence du type $\mathcal{L}$, Jan Kisyński, Colloquium Mathematicae, 1960

but I haven't read that one.